What is the rotational kinetic energy of the earth? Assume the earth is a uniform sphere. Data for the earth can be found inside the back cover of the book.

Verified step by step guidance

1. Start by recalling the formula for rotational kinetic energy: $K=\frac{1}{2}I{\omega }^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.
2. For a uniform sphere, the moment of inertia is given by $I=\frac{2}{5}m{r}^2$, where $m$ is the mass of the sphere (Earth) and $r$ is its radius.
3. Substitute the mass and radius of the Earth into the moment of inertia formula. The mass of the Earth is approximately $5.97\times {10}^{24}$ kg, and the radius is approximately $6.37\times {10}^6$ m.
4. Determine the angular velocity of the Earth. The Earth completes one rotation in approximately 24 hours. Convert this period into seconds and use the formula $\omega =\frac{2}{\pi }T$, where $T$ is the period of rotation.
5. Finally, substitute the values of $I$ and $\omega$ into the rotational kinetic energy formula to calculate the result.

Key Concepts

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation. It is calculated using the formula KE_rot = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity. For a uniform sphere, the moment of inertia can be determined using I = 2/5 m r², where m is the mass and r is the radius of the sphere.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For a uniform sphere, the moment of inertia is given by I = 2/5 m r², which reflects how the mass is distributed throughout the sphere, affecting its rotational dynamics.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object around an axis. It is typically measured in radians per second. For the Earth, angular velocity can be calculated based on its rotation period, which is approximately 24 hours for a complete rotation, allowing us to determine how quickly it spins on its axis.